Proactive customer service: operational benefits and economic frictions

Transcription

1 Proactive customer service: operational benefits and economic frictions Kraig Delana Nicos Savva Tolga Tezcan London Business School, Regent s Park, London NW1 4SA, UK kdelana@london.edu nsavva@london.edu ttezcan@london.edu Problem Definition: We study a service setting where the provider may have advance information about customers future service needs and may initiate service for such customers proactively if they are flexible with respect to the timing of service delivery. Academic / Practical Relevance: Information about future customer service needs is becoming increasingly available due to better system integration coupled with advanced analytics and big-data methods. We contribute to the literature by presenting a systematic analysis of proactive service as a tool that can be used to better match service supply with demand. Methodology: To study this setting, we combine i queueing theory, and in particular a diffusion approximation developed specifically for this problem, to quantify the impact of proactive service on customer delays with ii game theory to investigate economic frictions in a system with proactive service. Results: We find that proactive service reduces average delays, which we quantify with a closed-form approximation. More specifically, we show that this reduction is increasing concave in the proportion of customers who can be served proactively. Nevertheless, customers might not benefit from proactive service due to economic frictions; in equilibrium more customers will join the system and fewer will be willing to be flexible compared to social optimum. This is due to a positive externality leading to free-riding behavior customers who agree to be served proactively reduce waiting time for everyone else including those customers who do not agree to be served proactively. Managerial Implications: Our results suggest that proactive service may have a large operational benefit, but caution that it may fail to fulfil its potential due to customer self-interested behavior. Key words : Proactive Service, Queueing Theory, Game Theory, Healthcare Operations History : Revised: June 9, Introduction In many service settings e.g., healthcare, demand is highly variable but capacity is relatively fixed over short periods of time, leading to delays for customers. To reduce such delays, service providers often implement mechanisms that aim to modulate customer demand. These include providing delay information to discourage customers from joining the system when congestion is high Armony et al. 2009, Jouini et al. 2011, Ibrahim et al. 2016, Cui et al. 2014, offering customers the option to wait off-line or receive a call back Kostami and Ward 2009, Armony and Maglaras 2004a,b, or offering customers priority if they arrive during pre-allotted times De Lange 1

2 2 Proactive Customer Service et al In this paper, we investigate an alternative demand-modulation mechanism, proactive customer service, where the provider exploits information about customers future service needs to proactively initiate service when there is idle server capacity. Proactive service may find application in any a number of service systems. For example, this work was motivated by a healthcare setting, induction of labor a procedure where the process of delivering a baby is started artificially induced through the use of pharmaceuticals in a hospital ward. In this setting, the schedule of pre-planned and elective patients provided a list of customers with future service needs that the hospital could serve proactively see Appendix C for more details and a numerical illustration of this setting. Even in cases without a schedule, information on future customers may become available from other sources such as predictive models. For example, Jerath et al develop a method to predict which customers are likely to contact a health insurance call center based on claims data. The authors go on to suggest that a potential application would be to make advance outgoing calls to customers who have a high probability of calling, in other words, proactive customer service. The first contribution of this paper is to formulate a model that captures the benefits associated with proactive service and to develop novel approximations that quantify the improvement in system performance. The model is based on a Markovian queueing system with two queues in tandem. Arrivals to the first queue, which we call the orbit, represent virtual arrivals, or to be more exact, arrival of information about customers future service needs that the provider may choose to fulfill proactively. Because these customers are willing to receive service at a moment chosen by the provider, we find it convenient to label them as flexible. We assume the provider has minimal information about such flexible customers, having knowledge only of who requires service in the future but not when they would arrive. If a customer in orbit has not been served proactively, after a random amount of time they will transition to the second queue, which we label as the service queue. The service queue also experiences direct arrivals by customers who are not flexible, or equivalently, for whom the service provider does not have advance information about their service needs. Customers at the service queue are served in a first-come, first-served fashion irrespective of whether they arrived directly or through orbit by a single server. Naturally, proactive service only takes place if the server is idle i.e., the service queue is empty and there are customers in orbit. Using this queueing model, we show that proactive service reduces service queue congestion in the first-order stochastic sense proactive service exploits periods of idle capacity to bring forward arrivals who would have otherwise occurred at some point in the future. By doing so, proactive service smooths demand and, as a result, reduces delays for all customers, including those who are not flexible. Using a path-wise coupling argument, we show useful monotonicity results the greater

3 Proactive Customer Service 3 the proportion of flexible customers and the earlier the provider knows about their service needs the lower the average congestion and delay in the service queue. Finally, we develop a diffusion approximation that allows us to derive closed-form expressions for the average steady-state waiting times. The approximation suggests that the reduction in delay associated with proactive service displays decreasing marginal returns in the proportion of flexible customers. While the operational benefits associated with proactive service may be substantial, realizing them depends critically on customer behavior. On one hand, customers may refuse to be flexible, especially if there is an inconvenience cost associated with flexibility. On the other hand, the presence of proactive service, which reduces waiting times, may encourage customers to over-join the system compared to profit maximizing or social optimal, thus eroding any of the associated benefits. Furthermore, the benefit of proactive service for each individual customer will depend on what other customers do, i.e., it is an equilibrium outcome. Therefore, to understand whether proactive service will indeed be beneficial requires a game-theoretic analysis. The second contribution of this paper is to develop such a game-theoretic model to analyze customer behavior. To do so, we augment the standard to queue or not to queue dilemma Hassin and Haviv 2003 with the additional option to join the queueing system but be flexible. The game theoretic analysis identifies two economic frictions. First, customers will under-adopt proactive service compared to the profit maximizing or social optimum. This result is driven by a positive externality which gives rise to free-riding behavior: a customer who agrees to be flexible will reduce the expected waiting time of everyone else but this is a benefit that she does not take into account when making her own decision. In fact, we find instances where this economic friction can be extreme in the sense that a profit-maximizing provider or a central planner would have wanted all of the customers to be flexible, but in equilibrium, no customer chooses to be so. Second, we find that given the option to be served proactively, customers will over-join the system compared to both the profit maximizing or social optimal joining rate, as well as compared to a system without proactive service. This is due to the well-known negative congestion-based externalities e.g., Naor 1969 that proactive service exacerbates, i.e., for a given level of arrivals, proactive service reduces waiting times and, as a result, more customers would want to join compared to the case without proactive service. Interestingly, we find some surprising interactions between the positive and negative externalities. For example, an increase in the cost per unit of waiting time may lead to more customers joining the system. This is because the higher cost of waiting in the queue induces more customers to be flexible, which reduces waiting times, which in turn induces more customers to join the system. We conclude the paper by presenting two extensions of the queueing model described above. The first covers the multiserver case and shows that the basic intuition and approximations developed for

4 4 Proactive Customer Service the single-server case continue to hold in the multiserver case with minor modifications. The second extension examines the case where the information about future customer needs is imperfect. In this case, some of the customers served proactively did not require service. These errors could occur if customers may have their service need resolved through alternative channels e.g., spontaneous labor, or because of errors in information systems or predictive models in determining customers with future service needs. We show that the diffusion-limit approximation we developed for the case without errors can be adapted to derive closed-form approximations of system performance in the presence of errors. Using this approximation, we derive conditions under which proactive service reduces waiting times despite errors, and show that for some model parameters, the system can handle more errors as system utilization increases. This seems counter-intuitive at first because in a more heavily utilized system one would expect errors to increase delays more than in a less utilized system. However, this can be explained by the fact that reduction in delays gained through proactive service grows as utilization increases. Sketches of all proofs are presented in the Appendix of this paper. In the electronic companion EC we present detailed proofs, additional numerical/simulation analysis, and a numerical/simulation study with parameters calibrated to the induction of labor setting which motivated this work. This study suggests that proactive service can reduce average delays by up to 28% in this setting. Nevertheless, our analysis also suggests that the hospital should proceed carefully before implementing proactive service as economic frictions may lead to suboptimal voluntary adoption of the service. 2. Literature Review The analysis of proactive service in this paper contributes to three streams of queueing literature which are connected by the objective of better matching service supply and demand. The first stream examines interventions that modulate service supply in response to an exogenous demand process. The second stream focuses on interventions that seek to actively manage endogenous demand by taking into account the economic incentives of strategic customers. The third stream builds on the first two by incorporating future demand information. The first stream of literature considers supply-side interventions, e.g,. optimizing the number of servers, in response to exogenous changes in demand. The bulk of this literature is developed for call centers see Gans et al. 2003, Aksin et al for overviews and has focused on topics ranging from long-term workforce-management planning Gans and Zhou 2002, to medium-term shift staffing Whitt 2006, down to short-term call routing policies Gans and Zhou 2007, as well as combinations of short and medium-term solutions Gurvich et al Our work fits with the short-term strategies but, unlike the above-mentioned work, we assume that both system capacity

5 Proactive Customer Service 5 and the routing policy are fixed. One supply-side strategy that is closely related to proactive customer service is for idle servers to work on auxiliary tasks, such as s in call centers see, e.g., Gans and Zhou 2003 and Legros et al In the case of proactive service, future customers can be thought of as the auxiliary tasks, however, this substantially changes the dynamics of the system by smoothing the demand process. The second stream considers demand-side interventions that aim to influence strategic customers endogenous decisions. See Hassin and Haviv 2003 and Hassin 2016 for a comprehensive review of the economics of queues and strategic customer decision-making. One important intervention is the use of pricing to control the overall level of demand. What makes pricing particularly important in service systems is a key observation, first made by Naor 1969, that utility-maximizing customers tend to over-utilize queueing systems compared to the socially optimal level. This is due to customers imposing a negative externality on each other in the form of delays, and as a consequence, the service provider can increase welfare by charging customers a toll for joining the system. This finding persists in multiple variants, e.g., when the queue is unobservable Edelson and Hilderbrand 1975, and when customers are heterogenous or have multiple classes Littlechild 1974, Mendelson and Whang Naturally, the negative externality and over-joining persists in the presence of proactive service. However, in this setting we also find a rare instance of a positive externality, where customers who agree to be flexible reduce the waiting time of everyone else. 1 Beyond pricing, two other common demand-side interventions are delay announcements and multiple service priorities. Delay announcements encourage balking Allon and Bassamboo 2011, Armony et al. 2009, Ibrahim et al. 2016, Jouini et al or retrials Cui et al. 2014, especially when the system is congested. Multiple service priorities encourage some customers to wait in lowpriority queues usually offline, thus reducing the waiting time of high priority customers Engel and Hassin 2017, Armony and Maglaras 2004a,b, Kostami and Ward Our work is closer to the latter as one may think of customers who may be served proactively as arriving to a low priority queue. However, in contrast to the extant work, customers in this low priority queue may transition to the service system at any time, thus complicating the system dynamics. The third stream of literature to which our work is related focuses on the setting where the provider has information about the future. The benefits of future or advance demand information on production and inventory systems often modelled using queues has been recognized by many 1 We note that positive externalities are relatively rare in the literature of queueing games Hassin 2016, 1.8. Three notable exceptions are: i Engel and Hassin 2017, where customers that choose to join a low-priority queue reduce delays for customers that join the high-priority queue; ii Nageswaran and Scheller-Wolf 2016, where allowing one class of customers to wait in multiple queues may, under some conditions, reduce waiting time for customers who are only able to wait in a single queue; iii Hassin and Roet-Green 2011, where customers that pay to inspect the queue before making the decision to join or balk reduce waiting time for customers who do not inspect.

6 6 Proactive Customer Service e.g., Gallego and Özer 2001, Özer and Wei 2004, Papier and Thonemann More relevant is the work that considers customers who may accept product delivery early, i.e., are flexible to the timing of product delivery Karaesmen et al. 2004, Wang and Toktay The main difference between this stream of work and ours is that production and inventory systems largely focus on different performance measures e.g., cost of production, inventory cost, or stock-out costs as opposed to waiting times, and largely treat demand as exogenous. The study of future information in the context of service as opposed to inventory systems is more limited and has focused mainly on demand-side interventions in the form of admission control e.g, Spencer et al. 2014, Xu 2015, Xu and Chan As far as we are aware, the only other work that studies proactive service is Zhang This work was motivated by computing applications e.g., cache pre-loading or command pre-fetching and differs from ours in a number of dimensions. We present a more detailed comparison once we introduce our model in Operational Analysis: Single-server Queueing Model In this section, we present and analyze the proactive service system assuming there is a single server. The analysis has two goals: i to show that proactive service improves system performance, and ii to provide closed-form approximations that quantify the impact of proactive service on time-average measures of system performance Queueing Model We assume that demand arrives to the system following a Poisson process with rate λ, and that there exists two types of customers who require service, flexible and inflexible. The service times for both types of customers are assumed to be i.i.d. and exponential with parameter ; note we assume λ < throughout for stability. Inflexible customers make up a proportion 1 p of total demand and arrive to the service queue according to a Poisson process with rate 1 pλ. Upon arrival they immediately begin service if the server is free or join the queue, which operates in a first-in, first-out manner. For flexible customers, we assume the service provider becomes aware of the customer s service need some time before they actually arrive to the service queue and the provider has the option of serving them proactively at any time after becoming aware of their service need. To capture this, we assume that flexible customers do not arrive directly to the service queue, but instead arrive to a virtual queue, which we refer to as orbit. We assume arrivals to orbit follow a Poisson process with rate pλ. While in orbit customers may be served proactively if the server becomes idle, or, after a random period of time, which we assume to be i.i.d. and exponential with parameter > 0, they depart for the service queue on their own. Once at the service queue, flexible customers are served as any other customer who has arrived to the service queue directly. Together, these assumptions imply that the system may be modeled as two

7 Proactive Customer Service 7 Markovian queues in tandem linked by the proactive service mechanism, as depicted in Figure 1. We note that some of our results hold for more general time-in-orbit and service-time distributions. We indicate if this is the case when we state these results throughout the paper. We will refer to the parameter p as the proportion of flexible customers or, interchangeably, as the proportion of customers who have adopted proactive service. The average time flexible customers spend in orbit before transiting to the service queue on their own i.e., 1 / can be interpreted as the information lead time for flexible customers this is the average time in advance the provider knows of a customer s service need before the customer arrives to the service queue. We denote the occupancy of the orbit and the service queues at time t > 0 with N r t and N s t, respectively. Similarly, we denote the steady-state average occupancy and steady-state distribution of the queue length processes where they exist with N r, N s, and π = π r, π s, respectively. Finally, we define the steady-state average time for each customer type a {r, s}, spent in each queue b {r, s}, with T ab, if this exists. For example, Trs denotes the average time flexible customers spend in the service queue. We use the convention that a customer is assumed to be in the service queue while in service. Figure 1 Queueing model pλ 1 pλ Orbit Service queue 3.2. Impact of Proactive Service In order to assess the impact of proactive service, we compare the system with proactive service to a benchmark system without this capability, all other things being equal. In the benchmark case, the whole system can be modeled as a Jackson network where orbit is an M/M/ queue, the service queue is an M/M/1 queue, and all customers in orbit transition to the service queue. The steady-state distribution of queue lengths and waiting times for this system can easily be found in closed form Kleinrock 1976, see 3.2 & 4.4. The steady-state distribution of total number of customers in the service queue follows the geometric distribution with parameter 1 ρ where

8 8 Proactive Customer Service ρ := λ / < 1, and the steady-state distribution of the number of customers in orbit is Poisson with parameter pλ /. To denote the time average performance measures associated with the benchmark B system, we append superscript B to all the terms defined above; for example, N s denotes the expected number of customers in the service queue in steady state for the benchmark case. Impact of proactive service on queue lengths. We begin with the following result. Lemma 1. In steady state, the total number of customers in the system with proactive service is equal in distribution to the number of customers in the service queue without proactive service, that is, π r + π d s = πs B. Lemma 1 shows that the steady-state distribution of the total number of customers in the system with proactive service that is the sum of customers in orbit and in the service queue is equivalent to the steady-state distribution of number of customers in the service queue when proactive service is not possible. Interestingly, this implies that the distribution of the total number of customers in the system does not depend on the proportion of customers that is flexible i.e., p or the average information lead time i.e., 1 /. This result immediately implies that the average total occupancy in the system with proactive service equals the average occupancy of the service queue in the benchmark case i.e., Nr + N s = N s B = ρ /1 ρ. Furthermore, the non-negativity of the number of customers in orbit suggests that there is a stochastic ordering in the number of customers in the service queue, a result we present in Proposition 1. Throughout, we use to denote stochastic ordering. Proposition 1. newline i The steady-state distribution of the occupancy of orbit in the system with proactive service is stochastically dominated by that of the system without proactive service: π r πr B. ii The steady-state distribution of the occupancy of the service queue in the system with proactive service is stochastically dominated by that of the system without proactive service: π s πs B. The first part of the proposition establishes that the orbit is less occupied in a stochastic sense in the system with proactive service. This is not surprising. Since some customers are pulled from orbit to be served proactively, the time they spend in orbit is reduced and thus orbit becomes less congested compared to the system where proactive service is not possible. The second part of the proposition shows that the service queue is also less congested in a stochastic sense in the system with proactive service. Obviously, each part further implies that the time-average occupancy in both orbit and the service queue is reduced, that is, Nr N r B and N s N s B. We note here that Lemma 1 and Proposition 1 can be extended to the cases when time in orbit and/or service times are generally distributed.

9 Proactive Customer Service 9 Impact of proactive service on wait times. Because customers and service providers generally consider delay times and not system occupancy as the key metric of system performance, we now focus on the impact of proactive service on the expected time spent by each customer type in different parts of the system in steady state. The main result is provided in Proposition 2 and relies on Proposition 1 and the mean value approach MVA Adan and Resing 2002, 7.6. Proposition 2. Proactive service reduces delays for all customers in expectation: i T rr T B rr, ii T ss T B ss, iii T rs T B rs, but more so for those customers who can be served proactively: iv T B rs T rs T B ss T ss. The difference in expected time spent by flexible vs. inflexible customers in the service queue is proportional to the expected time spent in orbit: T ss T rs = λ T rr 0. 1 Proposition 2 shows that proactive service benefits both flexible and inflexible customers. The fact that proactive service benefits flexible customers is not surprising since some of them will be served proactively and receive service without having to wait in the service queue at all, proactive service will reduce the average waiting time for this class of customers. What is perhaps a little more surprising is that proactive service reduces waiting times for inflexible customers as well. This occurs because proactive service smooths demand by utilizing idle time to serve some customers early, thus, it reduces the likelihood that customers will arrive to a congested service queue. This reduction in congestion benefits all customers. However, Proposition 2 further implies that the benefit of proactive service is greater for flexible than inflexible customers. Impact of flexibility and information lead time. So far we have shown proactive service decreases occupancy in both orbit and the service queue as well as average delays for all customers when compared to a benchmark system without proactive service. Next, we establish a partial answer to the question of how the performance of a system with proactive service changes as the proportion of flexible customers and the information lead time change in Proposition 3. Proposition 3. newline i The steady-state distribution of number of customers in orbit i.e., π r is, in a stochastic ordering sense, increasing in p and decreasing in. ii The steady-state distribution of number of customers in the service queue i.e., π s is, in a stochastic ordering sense, decreasing in p and increasing in.

10 10 Proactive Customer Service Table 1 Monotonic behavior of performance measures N r Ns Trr Trs Tss p?? The arrow denotes that a given performance measure is increasing decreasing in p or. iii The performance measures exhibit the monotonic behaviors summarized in Table 1. Proposition 3 relies on a pathwise coupling argument to show part i, specifically that there are more customers in orbit in a stochastic sense in steady state if a larger proportion of customers are flexible and fewer are in orbit if there is shorter information lead time. Combining this result with Lemma 1 immediately implies the opposite impact on the service queue, which is given in part ii. Together these results imply the monotonicity of performance measures presented in part iii: that more information lead time i.e., smaller reduces time in the service queue for both flexible and inflexible customers, and that a greater proportion of flexible customers i.e., larger p leads to greater occupancy of orbit and lower occupancy of the service queue. We note that it is not possible to use the MVA approach to derive monotonicity results for the waiting times of flexible customers with respect to the proportion of flexible customers p. Therefore, we defer this to the next section where we develop diffusion limit approximations Approximations Based on Diffusion Limits In this section we present approximations based on diffusion limits for the performance measures we discussed in the previous section. To provide some intuition, in the diffusion limit, the primitive stochastic processes e.g., arrivals and service completions are replaced with appropriate limiting versions that make the occupancy processes more amenable to analysis. This enables the study of the macro-level behavior of the system over long periods of time and provides useful insights that are helpful in developing closed-form approximations of steady-state behavior Chen and Yao To proceed we need to define some additional notation. We focus on the system with proactive service see Figure 1 and we define a sequence λ n = c n for some c 0 and a sequence of systems indexed by n with these arrival rates. We still assume that arrivals are flexible with probability p and the departure rate from the service queue is, but we let the departure rate of each customer from orbit to the service queue be n = n. We further denote the number of customers in orbit and the service queue at time t as N n r t and N n s t, respectively. 2 We note that the simpler case, where customers never transition from orbit to the service queue on their own i.e., = 0, has been recently studied in Engel and Hassin In this case, the orbit becomes a low priority queue and the steady-state performance of the system can be obtained in closed form using exact analysis.

11 Proactive Customer Service 11 Asymptotic analysis. Observe that as n increases, the total arrival rate λ n approaches the service rate, which in turn implies that utilization goes to one. The part that is exploited by a diffusion limit is that utilization, and hence occupancy, grows at a specific rate. Knowing that the average number of customers in the n th system is λn / λ n is O n means that dividing through by n prevents the limit of the total occupancy process from going to infinity and hence the limits of both the orbit and service queue occupancy processes as well. We further scale time by replacing t by nt; this can be interpreted as the occupancy processes being observed over longer lengths of time as utilization approaches one to capture the macro-level behavior of the system. This leads to scaled occupancy processes ˆN r t = N n r nt / n and ˆN s t = N n s nt / n. Defining N n Qt = N n r t + N n s t 1 + to be the total number of customers in the system but not in service at time t, then the asymptotic behavior of the scaled queue processes ˆN n Qt = Nn Q nt / n is given by Theorem 1 below. Theorem 1. Assume that ˆN r n 0 = ˆN n Q0 pλn. For any finite T > 0, sup 0 t T ˆN n r t ˆN Qt n pλn 0 in probability as n. Theorem 1, which relies on the Functional Strong Law of Large Numbers and the Functional Central Limit Theorem Chen and Yao 2013, has a simple intuitive meaning. If the total scaled number of customers in the system excluding those receiving service, ˆN n Q t, is less than pλn /, then there are almost no customers waiting to receive service in the service queue more precisely it is o n; alternatively if the total is greater than pλn /, then the scaled number in orbit is pλn / and almost all others are in the service queue. More generally, Theorem 1 implies that, given the total number of customers in the limiting system, we now know how they are distributed between the orbit and the service queue. In other words, the state space collapses. The state-space collapse result is similar to Proposition 3.1 in Armony and Maglaras 2004b, where the service provider offers customers call backs with a service guarantee. In their setting, customers who agree to receive a call back are also placed in a holding system akin to orbit in our setting. However, the driving mechanism and the proof techniques are significantly different. In our setting the orbit queue functions similarly to a low-priority queue in that if there are customers in the service queue, they are served exclusively; therefore, the service queue empties out faster than orbit. In contrast, in the setting of Armony and Maglaras 2004b, customers in orbit are sometimes given priority over the customers in the service queue this happens when the number of customers in orbit exceeds the limit pλn otherwise the system would not meet the call-back guarantee. The diffusion limit presented above is also related to those developed in the queueing literature with abandonments; see Ward and Glynn 2003 and Borst et al The main

12 12 Proactive Customer Service difference in our model is that customers do not abandon but transition from orbit to the service queue. Therefore we need to use a different scaling to obtain meaningful limits approximations. For instance, if we used the scaling in Theorem 1.1. in Ward and Glynn 2003, the service queue would always be asymptotically empty, which does not lead to useful approximations. Hence we use an alternative scaling where the transition rate from orbit to the service queue scales at a faster rate; more specifically, it scales at the same rate as the utilization of the system. This scaling, however, introduces a technical difficulty because the orbit occupancy can change rapidly even in the limit. Nevertheless, we are able to prove that there is a state-space collapse in the limit, which leads to the diffusion limit presented above. Approximations. In order to develop closed-form approximations for system performance, the next step is to apply the asymptotic result on the allocation of customers between the orbit and the service queue to a finite system. Since the exact results show that the total number of customers in the system is distributed geometrically, we apply the split of customers implied by Theorem 1 assuming it holds for finite n. Computing the expected value of the occupancy of the service queue yields the following approximations, where N s ρ + ρ pλ pλ + 1 pλ + ρ 1 ρ x denotes the floor function. The second approximation follows from ρ 1 ρ 1 ρ pλ, 2 1 ρ pλ pλ. Utilizing MVA see also Proposition 2, the approximation given by 2 can be used to estimate all other performance measures for queue lengths and wait times. By the PASTA property, the memoryless property of service times, and 2, the average time spent in the service queue for inflexible customers is T ss = N s By 2 and the implication of Lemma 1 that N s + N r = N r = ρ 1 ρ 1 ρ pλ ρ ρ 1 ρ N s ρ 1 ρ ρ 1 ρ pλ ρ, we have that, 1 ρ By Little s Law and 4, we have that the average time spent in orbit is. 4 T rr = N r pλ 1 p λ ρ 1 ρ pλ, 5 and finally by equation 1 and the approximations 5 and 3, we can find an approximation of the average time spent in the service queue for flexible customers T rs. The approximation given by equation 2 for the average number of customers in the service queue has an intuitive appeal. It is equal to the average number of customers at the service queue

13 Proactive Customer Service 13 in the absence of proactive service ρ /1 ρ, multiplied by a constant, 1 ρ1 ρ pλ/ 1, that represents the benefit of proactive service. As expected, this benefit disappears i.e., the constant goes to one if there are no flexible customers i.e., p = 0 or the average information lead time goes to zero i.e., 1/ 0. Furthermore, the approximations above allow us to derive additional properties of performance measures that could not be derived using exact analysis see Table 1. For instance, using equation 2, we can show that the service queue occupancy decreases exponentially with p /, which implies there are decreasing marginal benefits in the proportion of customers that are flexible and the average information lead time. In addition, using equation 5 we can show that T rr is monotonic decreasing in p. Also T rs is monotonic decreasing in p provided λ and ρ >.2. Verifying the accuracy of the approximations: Because the approximations presented above are based on an asymptotic result, in this section we examine their accuracy in finite systems where utilization is less than one. For instance, Figure 2 depicts the comparison of the diffusion approximations and simulated average delays for the case when λ = 0.875, =.2, and = 1. A full sensitivity analysis of the accuracy of the approximations is given in Appendix B.1.1. In general, the approximations perform remarkably well for all values of p when utilization is high i.e., ρ.75, 1, and information lead time is not too large i.e., λ / 1. This is not surprising given the asymptotic regime deployed to develop the approximations assumed that λ n 0 at the same rate as n 0. 3 Figure 2 also serves to illustrate the substantial reduction in delays derived from proactive service. For instance, if all customers are flexible i.e., p = 1 the total average delay in the service queue is reduced by 38.7% from 8.0 to 4.90 time units. Even if only half of customers are flexible i.e., p = 0.5, the average time in the service queue is reduced by 22.2% from 8.0 to 6.23 time units. This reduction in delays is achieved even though the average information lead time is relatively short only 62.5% of the expected delay in the benchmark case. In other words, relatively little information lead time goes a long way when customers can be served proactively. Remark 1 When time in orbit is a constant. Although not essential for the rest of our analysis, we compare the reduction in wait times achieved with proactive service when time in orbit is exponentially distributed, to the case when time in orbit is deterministic. The latter case is studied in Zhang 2014 and can be interpreted as a case when the provider possesses more information about the customers future service needs compared to the former. The setting in Zhang 2014 has two additional differences. First, it assumes that a customer does not have to be present 3 As the diffusion limit presented above fails when the information lead time increases i.e., 1/, not surprisingly, the approximation does not collapse to the exact analysis presented in Engel and Hassin 2017 where the authors assume that = 0. Therefore, the two results can be seen as applicable to different parameter regions.

14 14 Proactive Customer Service Figure 2 Customer delays in the proportion of flexible customers p, where λ =.875, =.2, = 1. 8 T ss 7 T rs Units of time T rr Diffusion approximation Simulation Proportion of flexible customers p for service. Hence, the waiting time measure they consider does not include service time, only time in queue. It is straightforward to modify their approach to include service time as well. This is the approximation we present here. Second, Zhang 2014 assumes that inflexible customers have preemptive priority over flexible customers. This assumption is essential for his analysis technique, however, it is not realistic for service systems. Hence, we only compare our results to his when p = 1, in which case preemptive priority does not matter because there are no inflexible customers. Let w denote the time customers spent in orbit before they transition to the service queue. Under this assumption, Zhang 2014 shows that the average amount of time customers spend in the service q queue excluding time spend in service when p = 1 is T ss = ρ λ e 1 ρw. Based on 2, with p = 1 q and = 1/w, we arrive at the following approximation, T ss = ρ λ ρλw q. Let ρ = T ss / T ss, q then ρw. we have ρ = e w e ρ It can be shown that limρ 0 ρ = 0, lim ρ 1 ρ = 1, and that is strictly increasing in ρ. Therefore, knowing exactly when customers would transition from the orbit to the service queue helps further reduce average time spent in service queue. However, this additional reduction in waiting time decreases as the system reaches heavy traffic. 4. Economic Analysis: Endogenous Decision-Making and Welfare The queueing analysis thus far has shown the significant potential of proactive service to improve operational performance. However, it assumes exogenous customer arrival rates to both orbit and service queue. This is unlikely to be realistic in many service settings because it is customers who choose to join the queue and/or to be flexible based on the costs and benefits of each option.

15 Proactive Customer Service 15 Therefore, to understand the benefits of proactive service, we need to consider the decisions of strategic customers in equilibrium. To do so we build off a standard queueing game e.g., Hassin and Haviv 2003 where, in addition to the option of joining or not, customers need to also choose if they accept to be flexible. In such a system, customer self-interested behavior generates two distinct economic frictions. The first has to do with the customer joining decision any customer who joins the system increases the waiting time for everyone else. This is a negative externality customers do not take into account when deciding to join the system, leading to customers over-utilizing the system compared to the social or profit maximizing optimal Naor The option to be flexible introduces a second friction. Any customer that chooses to be flexible reduces the waiting time for everyone else. This is a positive externality that customers do not take into account when deciding whether to be flexible or not, leading to customers under-adopting proactive services. Moreover, as we show in the next section, these two opposing externalities interact in non-trivial ways Customer Utility and Equilibrium Demand To facilitate a game theoretic analysis, we assume that there exists a large population of potential customers who are homogeneous, rational, and risk-neutral economic agents. We also assume that customer waiting times are accurately approximated by the smooth version of the closed-form diffusion approximations of 3. Each customer has some small exogenous probability of requiring service such that, in aggregate, customer service needs can be modelled by a Poisson process with rate Λ. Receiving service is valued at v and each customer also has access to an outside service option whose value we normalize to zero. Customers decide whether to join, and if they join whether to be flexible, by examining the expected cost of these choices which we assume is common knowledge. More specifically, we assume that real-time waiting time information is not available but customers have an accurate belief about average waiting times; see Chapter 3 of Hassin and Haviv 2003 for an extensive review of the theory and applications of unobservable queues. The expected costs have three sources. First, all customers are averse to waiting at the service queue and incur a waiting-time cost w s 0 per unit of time spent there waiting or receiving service. Second, flexible customers need to be ready to answer the call from the idle service provider at any time and therefore incur i an opportunity cost 0 w r w s per unit time spent in orbit that reflects any inconvenience associated with waiting to commence service early; ii a fixed inconvenience cost h 0, which can be interpreted as the cost of giving up autonomy/spontaneity in the timing of joining the queue. Third, customers may need to pay prices c r 0 and c s 0 set by the provider for flexible and inflexible customers, respectively. Given the assumptions, the expected utility of customers who choose to join but

16 16 Proactive Customer Service not to be flexible is v c s w s Tss, the expected utility of customers who join and are flexible is v c r h w r Trr w s Trs, and the utility of customers who do not join is zero. Customers choose to 1 not join, 2 join and be flexible, or 3 join and be inflexible, based on the option with the greatest expected utility. For notational convenience, we let λ Λ represent the effective demand i.e., arrival rate to the system such that J = λ /Λ [0, 1] gives the proportion of customers who join the system, and p [0, 1] represents the proportion of customers who choose to be flexible conditional on joining. Because customers are homogeneous, we are interested in symmetric Nash Equilibria where, given that all other customers play a mixed strategy represented by J, p i.e., join with probability J and are flexible with probability p, each customer s best response is to also play strategy J, p. For the rest of the analysis we restrict our attention to λ rather than J as there is a one-to-one correspondence between the two Unregulated Customer Equilibrium To study the incentives introduced by proactive service, we examine the case where customers make their own utility-maximizing decisions in an unregulated system, i.e., where c s = c r = 0. Under mild assumptions, Proposition 4 establishes the existence and uniqueness of equilibrium as well as comparative statics. Proposition 4. If Λ ws ws.75, v 4 and, then: v i. There exists a unique symmetric Nash Equilibrium p e, λ e for customer flexibility and joining behavior. ii. The equilibrium strategy is such that: a The proportion of flexible customers p and the arrival rate λ e are non-increasing in the costs of flexibility h and w r. b The proportion of flexible customers p e is non-increasing in customer valuation v, and the arrival rate λ e is non-decreasing in customer valuation v. c The proportion of flexible customers p e is non-decreasing in the waiting-time cost w s, but the arrival rate λ e can be decreasing or increasing in the waiting-time cost w s. Specifically, if all strategies are played with positive probability so that λ e < Λ and p e 0, 1, then the arrival rate λ e is increasing in the waiting-time cost w s, otherwise λ e is decreasing in the waiting-time cost w s. The conditions under which this proposition holds also ensure that utilization is relatively high and information lead time is relatively low, therefore ensuring that the diffusion approximations are a good representation of the system performance. We prove Part i by construction, considering all possible cases and proving uniqueness and existence though enumeration. The comparative statics results in Part ii rely on the monotonicity of delays in both the arrival rate and the proportion

17 Proactive Customer Service 17 of flexible customers, and are largely in line with intuition. Part iia shows that, as the costs of flexibility h, w r increase, fewer customers agree to be flexible and fewer customers join, just as one might expect. Part iib shows that, as customer valuation for service v increases, more customers join, which is also as expected. More interestingly, Part iib also shows that, as customer valuation v increases, a smaller proportion of those who join choose to be flexible, which suggests a nonobvious interaction of externalities. Specifically, this happens because as congestion increases i.e., more customers join due to their valuations v being higher the value of free riding i.e., the value a customer gets when other customers choose to be flexible also increases. As a result, the proportion of customers who agree to be flexible becomes smaller. Perhaps even more surprising is Part iic, which shows that, as the cost of time spent in the service queue w s increases, the arrival rate may actually increase. The reason is that the increase in waiting-time cost w s induces more customers to be flexible, which generates a positive externality i.e., reduces average waiting time, and in turn induces more customers to join. In this way one can clearly see the positive externality associated with flexibility interacts with the negative externality associated with congestion Customer suboptimal behavior: Over-utilization and Free-Riding Next, we seek to understand how customer decisions in an unregulated equilibrium compare to those that a profit-maximizing service provider would want. The provider seeks to maximize the revenue rate from prices paid by customers subject to customers equilibrium behavior, 4 i.e., max c r,c s 0 λpc r + 1 pc s 6 subject to: p, λ is an equilibrium. Because customers are homogeneous, a profit maximizing provider will not find it optimal to set prices such that customers are left with a positive surplus in equilibrium; if this was the case the provider would be able to increase prices without impacting customer decisions Hassin and Haviv 2003, 1.3. Therefore, the profit maximizer will set prices such that c s = v w s Tss p e, λ e and c r = v h w r Trr p e, λ e w s Trs p e, λ e. Given this, the provider s revenue can be rewritten as, W p, λ = λ [ pv h w r Trr p, λ w s Trs p, λ + 1 pv w s Tss p, λ ]. 7 An interesting observation is that the profit of the provider in this case is equal to the average welfare rate of customers, an observation also made by Hassin and Haviv 2003, 1.3. In other words, 4 We note that, since proactive service may require a one-off cost to implement and perhaps a variable cost to monitor customer service needs, the provider will need to compare any increase in revenue to the implementation and runningcosts to determine whether or not to implement proactive service. However, since this is a straightforward comparison, we do not model this explicitly and assume proactive service can be implemented at zero cost. We will therefore use revenue and profit interchangeably.

18 18 Proactive Customer Service the profit maximizer would want customers to behave in exactly the same way as a benevolent social planner whose aim is to maximize welfare. The only difference is that the profit maximizer would set the prices so as to extract all of the customer surplus, while the social planner, for whom prices are an internal transfer, would be indifferent between any price. Therefore, to understand whether customers autonomous joining decisions of 4.2 are suboptimal for the profit maximizer, it would suffice to compare them to those of a benevolent social planner. The existence of the optimal solution to the social planner s problem is guaranteed by the fact that the action space is compact and the objective function is continuous. 5 We compare the socially optimal customer actions with the equilibrium customer decisions of Proposition of 4.2 in the next result. Proposition 5. If Λ.75, v 16 ws, w s/v, then for any socially optimal/profitmaximizing solution p so, λ so, i. Customers over-utilize the system compared to the socially optimal/profit-maximizing solution, λ so λ e. ii. Customers under-adopt proactive service compared to the socially optimal/profit-maximizing solution, p so p e. In particular, there exist thresholds of the flexibility cost h denoted by h and h, where 0 < h < h, such that if h h then p e = 0 and if h h then p so = 1. This implies that if h h h then p e = 0 and p so = 1, i.e., no customer would choose to be flexible in equilibrium but the social planner or profit maximizer would designate all customers who join to be flexible. The conditions under which this proposition holds are a subset of the conditions of Proposition 4 and, as was the case there, they also ensure that the diffusion approximations are a good representation of the system performance. Proposition 5 shows that customers will over-utilize a service system with proactive service and under-adopt proactive service the option to be flexible compared to the socially optimal or the profit-maximizing solution. Figure 3 illustrates this point by showing the equilibrium strategy and the socially optimal/profit-maximizing strategy as a function of the fixed cost of flexibility h for a specific example. As can be seen in Figure 3a, the underadoption of proactive service can be substantial in the sense that there exists a region 0.3 < h < 1.4 in Figure 3a, where the central planner would have dictated that all customers who join be flexible but in equilibrium flexibility is a strictly dominated strategy. Proposition 5 Part ii shows that such a region is not specific to this example but always exists. In this region, customers would be better off if they collectively chose to be flexible, but because customers are individually better off by 5 We note that we are unable to prove that the social planner s objective W p, λ is concave. Nevertheless, in numerical experiments we find the first order conditions are both necessary and sufficient.